Cubes in 4 dimensions.

Mathematicians often work with the cube, when describing
various physical situations, and indeed SOMA was first discovered
at a mathematical lecture of space divided in cubes.

One other interesting aspects of the cube is that it is
one of the objects of which we can glimpse the fourth
dimension.

We will probably never know if the fourth dimension is
anything else than the brainchild of mathematicians,
but investigating the properties of a fourth dimension
makes it a lot easyer for the scientists to describe the
way objects interact here in our 3-D world.

When scientists describe the world and the universe
in which we live, they often have to take
the fourth 'Space-dimension' into account. No one can
see this dimension, but just as you can fold a flat
two-dimensional piece of paper into a three-dimensional
cube, then the mathematicians can compute from our 3-D
world, into the 4-D so called 'hyperworld'

The mathematical concept of dimensions is actually
quite simple. A dot has no dimension because you
cannot move anywhere on it. A straight line has the
dimension 1. because you can move in one direction.-
(length wise)
Extending the line at a rightangled
direction gives us the sheet, like a piece of paper,
and here we have 2 dimensions.- (length and width)
Accordingly, we can extend the sheet in a direction
perpendicular to the sheet surface in order to get
the cube, now with 3 dimensions.- (length, width and
height)

Now, if we take this a step further. Extending the
cube in a direction perpendicular to ALL the existing
axes then we enter the hyper space having 4 dimensions.

Let us see how dimensions govern the evolution
of a 3-D cube, how we may view the shadow of a 4
dimensional cube, and how a 4 dimensional cube
will look when it is "unfolded" to our 3-D world.

A point has 1 terminal point (By definition).

Moving a point in a straight line produces a Line
with 2 terminal Points (Corners).

Moving the line along a straight path produces a
Square with 4 corners.

The numbers 1, 2, 4, are in a Geometrical Progression
where the next number is then 8.
And indeed, moving the square along a straight path
produces a cube with 8 corners.
A logic assumption is then that moving a cube along
a straight path will produce a hypercube with 16
corners. - And so it does.

From this we may deduce the numbers that a 4 dimensional
equivalent of a cube, will have:

Shape

Dimensions

Corners

Edges

Faces

Volumes

Dot

0

1

0

0

0

Line

1

(2)

2

0

0

Sheet

2

4

4

2

0

Cube

3

8

12

6

1

Hypercube

4

16

32

24

8

It is difficult for us to imagine that this should
be possible so let us back up for a short while,
and unfold the cube.

A 3-D cube consist of 6 squares of 2 dimensions.

We all recognice the familiar 'cross' shape, that
can be cut out of a sheet of 2-D paper and then be
folded back into a hollow cube.
Likewise it is possible to "unfold" a hollow 4
dimensional hyper cube.

A Hypercube also have a - sort of - surface,
consisting of 8 cubes of 3 dimensions.

So doing the unfold, we get a 3 dimensional structure
resembling a 4 armed cross.
Still - although difficult to imagine - this figure
can be cut from solid 3-D materials and folded back
into the 4-D hyper cube.

Now, if we look at the familiar 2-D cross, we can
recognise the middle piece as being the bottom of the
finished cube, and the 'tail square' as being the top
side of the cube.
In a quite similar fashion we must imagine the center
piece of the 3-D"cross" as being the "bottom" of
our hyper cube, and the "tail piece" as being the "top".

Although we cannot see this world of 4 dimensions,
mathematicians can tell us, that strange things can
be done there.
For example: Rotating a 3-D figure through a 4-D space
and back will produce a mirrored shape.
Just like you do, if you write your name on a piece of
flat glass, and then flip it through our 3-D space, to
get the underside upwards.
So sending your "left shoe" into the 4-D world,
rotating it, and getting it back, will produce a "right shoe"

In the 4-D universe the SOMA pieces #5 and #6 are no
longer different pieces. They are identical!To a 4-D person, SOMA pieces are as flat as TETRIS
pieces are to us.

On the chemical scale, sugar will taste like starch
and starch will taste like sugar.
Because the molecules are 3-D mirror's of each other.

Just as we can see the interior of a figure drawn
on a flat 2-D piece of paper, so a 4-D person can see
the interior of all our objects.
The 4-D person could remove anything from a closed
drawer, without opening it, simply because our drawer
is closed on ALL 3 sides, but not on the 4-D sides,
leaving free access to all contents.

Now, HOW would a 4-D cube look to us, if we were
standing in front of it.?

Three-dimensional slicing

To answer such a question, we must first examine
the consept of intersections.

Imagine a 2-D creature living on a flat surface.
and imagine further that a 3-D cube is about to fall
through his world.
Now. what would he see ?
Well. it depend on how the cube is orientated.!

If the cube falls flat on the surface, he would see
a square that appear and disappear.

If the cube falls on an edge, he would see a line
that would grow into a square, and thereafter diminish
and disappear.

Finally. If the cube falls on a corner, Like the
image to the right, he would see a spot, growing into
a triangle, then into a hexagon, back into a diminishing
triangle, and finally to disappear in a spot.
A strange experience this would be for the 2-D creature.
Click this figure to try the intersections yourself.

Four-dimensional slicing

Quite likewise our experience of a 4-D object will
be a 3 dimensional object of complex appearence.
By slicing our 4-D hypercube into the 3-D sections
that appear in our 3-D world, we can gain some insight
into the 4'th dimension.
It does require a LOT of imagination, because like
the 2-D creature should imagine our 3-D cube by viewing
its slices, so we must imagine an unseen 4-D figure,
but viewing it's 3-D slices.

Like it is for the 3-D cube, the orientation of
the hypercube does influence what we see.
In fact, the slices of the hypercube very much resemble
those of the cube. For example, if a hypercube passes
through our space head-on (starting with one of its eight
three-dimensional cubic faces) then it looks like a
series of cubes, all equal in size, just like slicing
the cube face first gave a series of squares.
Do you see the parallels?

Slicing the hypercube from a two-dimensional
side first gives resuls analogue to the cubic
slices from an edge.

Now visualize the shape of the hypercube by slicing
it from a vertexsee the figure to the right
(Imagine it, being a 4-D hypercube, suspended in a
rubber string, dangling up and down through our 3-D world.)
To try the slicing yourself, click below.

Click this figure to try the intersections yourself.

Fold-Outs

Slicing is a good way to understand shapes better
because it breaks them down into a series of
lower-dimensional objects. Another good way to understand
a shape is to try to build it from a lower-dimensional
"fold-out," like a cardboard box unfolded and placed flat
on the floor.

Three dimensional fold-outs

Consider this picture of an unfolded cube:

Does it look like a cube? Now, if we were
to give you instructions that said "connect the blue edges"
to build a cube? The best a 2-D person could do would be
to stretch the squares and align the blue edges together
in the plane.

But this was not what we were looking for.
We wanted them to fold up without stretching
the squares, something that requires three dimensions,
something they would not be able to do. The fold-out and
instructions for putting it together could nonetheless
help the 2-D people gain a better appreciation for
what a real cube is.

Given a third dimension to work with, the
foldout could actually create the cube by
connecting the blue edges.

Four-dimensional fold-outs

For us, who live in a 3-D world, we can gain
a better understanding of four-dimensional shapes
by examining their three-dimensional fold-outs.

Imagine a "cube" where 3 sides are missing, like
the drawing above - the minimum material required to
build such a cube, is three squares. When we talk about
the fourth dimension, the situation is analogue:
The minimum material required to build a 4-D shape is
three cubes arranged around an edge, and then connecting
the blue faces without stretching the cubes.

As 3-D persons we cannot complete this task.
The best we can do is to stretch the cubes to align
the blue faces.

But a four-dimensional creature could fold
the figure into a hypercube with no problem.

Here is the complete hypercube fold-out:

As mentioned before, notice here the
similarities of this fold-out to the cube fold-out.
We have a central object (square or cube) surrounded
on each face with other identical objects, and then a
final one stuck on the bottom.
It is easy to think of the center object as being
the "base" of the folded-up figure, and the extra
object on the end as being the "top" of the figure.

Now - Try to rotate a hypercube yourself.

One thing is to read a static text about a
subject, especially about such a graphically intensive
subject as 4 dimensions. Another thing is HANDS ON
experience.
To allow just that, Michael Gibbs has written a very
impressive Java Applet that shows 4-dimensional polytopes,
(Figure) and lets you rotate them.

Here is a theory of my friend 'Courtney McFarren'

When we talk about mathematics, my good friend
Courtney always starts talking - a lot - which is great,
and when the talk then turns 4 dimensional he has some
very interesting ideas (and I believe he is right)
Here's what he said:

If it wasn't for 4-D space, gravity would not exist. Our
universe is 3-D space wrapped around a huge 4-D hyper-sphere.
All matter is attracted to each other, so they push towards
each other and towards the center of the hyper-sphere,
leaving "space dents" around them. These dents (or pockets)
warp the space around any piece of matter, creating the
"hold" of gravity. The larger the piece of matter, the
deeper the "dent" becomes, therefore the stronger the gravity.

When we view a distant galaxy, not only are we
viewing as it was billions of years ago, but we are also
viewing it when the radius of the universe was much smaller.
Because of that, the light that reaches Earth had to travel
in a long "spiral" path, but not a short "arc" path, which
is probably why we think certain objects are farther away
and older than they really are.